Theorem sxbrsigalem0 30333

Description: The closed half-spaces of ( RR X. RR ) cover ( RR X. RR ). (Contributed by Thierry Arnoux, 11-Oct-2017.)

Assertion

Ref	Expression
sxbrsigalem0	|- U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) = ( RR X. RR )

Distinct variable group:   e, f

Proof of Theorem sxbrsigalem0

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unissb 4469	. . 3 |- ( U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) C_ ( RR X. RR ) <-> A. z e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) z C_ ( RR X. RR ) )
2		elun 3753	. . . 4 |- ( z e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) <-> ( z e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) \/ z e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
3		eqid 2622	. . . . . . . . 9 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) = ( e e. RR |-> ( ( e [,) +oo ) X. RR ) )
4	3	rnmptss 6392	. . . . . . . 8 |- ( A. e e. RR ( ( e [,) +oo ) X. RR ) e. ~P ( RR X. RR ) -> ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) C_ ~P ( RR X. RR ) )
5		pnfxr 10092	. . . . . . . . . . 11 |- +oo e. RR*
6		icossre 12254	. . . . . . . . . . 11 |- ( ( e e. RR /\ +oo e. RR* ) -> ( e [,) +oo ) C_ RR )
7	5, 6	mpan2 707	. . . . . . . . . 10 |- ( e e. RR -> ( e [,) +oo ) C_ RR )
8		xpss1 5228	. . . . . . . . . 10 |- ( ( e [,) +oo ) C_ RR -> ( ( e [,) +oo ) X. RR ) C_ ( RR X. RR ) )
9	7, 8	syl 17	. . . . . . . . 9 |- ( e e. RR -> ( ( e [,) +oo ) X. RR ) C_ ( RR X. RR ) )
10		ovex 6678	. . . . . . . . . . 11 |- ( e [,) +oo ) e. _V
11		reex 10027	. . . . . . . . . . 11 |- RR e. _V
12	10, 11	xpex 6962	. . . . . . . . . 10 |- ( ( e [,) +oo ) X. RR ) e. _V
13	12	elpw 4164	. . . . . . . . 9 |- ( ( ( e [,) +oo ) X. RR ) e. ~P ( RR X. RR ) <-> ( ( e [,) +oo ) X. RR ) C_ ( RR X. RR ) )
14	9, 13	sylibr 224	. . . . . . . 8 |- ( e e. RR -> ( ( e [,) +oo ) X. RR ) e. ~P ( RR X. RR ) )
15	4, 14	mprg 2926	. . . . . . 7 |- ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) C_ ~P ( RR X. RR )
16	15	sseli 3599	. . . . . 6 |- ( z e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) -> z e. ~P ( RR X. RR ) )
17	16	elpwid 4170	. . . . 5 |- ( z e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) -> z C_ ( RR X. RR ) )
18		eqid 2622	. . . . . . . . 9 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) = ( f e. RR |-> ( RR X. ( f [,) +oo ) ) )
19	18	rnmptss 6392	. . . . . . . 8 |- ( A. f e. RR ( RR X. ( f [,) +oo ) ) e. ~P ( RR X. RR ) -> ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) C_ ~P ( RR X. RR ) )
20		icossre 12254	. . . . . . . . . . 11 |- ( ( f e. RR /\ +oo e. RR* ) -> ( f [,) +oo ) C_ RR )
21	5, 20	mpan2 707	. . . . . . . . . 10 |- ( f e. RR -> ( f [,) +oo ) C_ RR )
22		xpss2 5229	. . . . . . . . . 10 |- ( ( f [,) +oo ) C_ RR -> ( RR X. ( f [,) +oo ) ) C_ ( RR X. RR ) )
23	21, 22	syl 17	. . . . . . . . 9 |- ( f e. RR -> ( RR X. ( f [,) +oo ) ) C_ ( RR X. RR ) )
24		ovex 6678	. . . . . . . . . . 11 |- ( f [,) +oo ) e. _V
25	11, 24	xpex 6962	. . . . . . . . . 10 |- ( RR X. ( f [,) +oo ) ) e. _V
26	25	elpw 4164	. . . . . . . . 9 |- ( ( RR X. ( f [,) +oo ) ) e. ~P ( RR X. RR ) <-> ( RR X. ( f [,) +oo ) ) C_ ( RR X. RR ) )
27	23, 26	sylibr 224	. . . . . . . 8 |- ( f e. RR -> ( RR X. ( f [,) +oo ) ) e. ~P ( RR X. RR ) )
28	19, 27	mprg 2926	. . . . . . 7 |- ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) C_ ~P ( RR X. RR )
29	28	sseli 3599	. . . . . 6 |- ( z e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) -> z e. ~P ( RR X. RR ) )
30	29	elpwid 4170	. . . . 5 |- ( z e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) -> z C_ ( RR X. RR ) )
31	17, 30	jaoi 394	. . . 4 |- ( ( z e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) \/ z e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) -> z C_ ( RR X. RR ) )
32	2, 31	sylbi 207	. . 3 |- ( z e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) -> z C_ ( RR X. RR ) )
33	1, 32	mprgbir 2927	. 2 |- U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) C_ ( RR X. RR )
34		funmpt 5926	. . . . . 6 |- Fun ( e e. RR |-> ( ( e [,) +oo ) X. RR ) )
35		rexr 10085	. . . . . . . . . . 11 |- ( ( 1st ` z ) e. RR -> ( 1st ` z ) e. RR* )
36	5	a1i 11	. . . . . . . . . . 11 |- ( ( 1st ` z ) e. RR -> +oo e. RR* )
37		ltpnf 11954	. . . . . . . . . . 11 |- ( ( 1st ` z ) e. RR -> ( 1st ` z ) < +oo )
38		lbico1 12228	. . . . . . . . . . 11 |- ( ( ( 1st ` z ) e. RR* /\ +oo e. RR* /\ ( 1st ` z ) < +oo ) -> ( 1st ` z ) e. ( ( 1st ` z ) [,) +oo ) )
39	35, 36, 37, 38	syl3anc 1326	. . . . . . . . . 10 |- ( ( 1st ` z ) e. RR -> ( 1st ` z ) e. ( ( 1st ` z ) [,) +oo ) )
40	39	anim1i 592	. . . . . . . . 9 |- ( ( ( 1st ` z ) e. RR /\ ( 2nd ` z ) e. RR ) -> ( ( 1st ` z ) e. ( ( 1st ` z ) [,) +oo ) /\ ( 2nd ` z ) e. RR ) )
41	40	anim2i 593	. . . . . . . 8 |- ( ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. RR /\ ( 2nd ` z ) e. RR ) ) -> ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. ( ( 1st ` z ) [,) +oo ) /\ ( 2nd ` z ) e. RR ) ) )
42		elxp7 7201	. . . . . . . 8 |- ( z e. ( RR X. RR ) <-> ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. RR /\ ( 2nd ` z ) e. RR ) ) )
43		elxp7 7201	. . . . . . . 8 |- ( z e. ( ( ( 1st ` z ) [,) +oo ) X. RR ) <-> ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. ( ( 1st ` z ) [,) +oo ) /\ ( 2nd ` z ) e. RR ) ) )
44	41, 42, 43	3imtr4i 281	. . . . . . 7 |- ( z e. ( RR X. RR ) -> z e. ( ( ( 1st ` z ) [,) +oo ) X. RR ) )
45		xp1st 7198	. . . . . . . 8 |- ( z e. ( RR X. RR ) -> ( 1st ` z ) e. RR )
46		oveq1 6657	. . . . . . . . . 10 |- ( e = ( 1st ` z ) -> ( e [,) +oo ) = ( ( 1st ` z ) [,) +oo ) )
47	46	xpeq1d 5138	. . . . . . . . 9 |- ( e = ( 1st ` z ) -> ( ( e [,) +oo ) X. RR ) = ( ( ( 1st ` z ) [,) +oo ) X. RR ) )
48		ovex 6678	. . . . . . . . . 10 |- ( ( 1st ` z ) [,) +oo ) e. _V
49	48, 11	xpex 6962	. . . . . . . . 9 |- ( ( ( 1st ` z ) [,) +oo ) X. RR ) e. _V
50	47, 3, 49	fvmpt 6282	. . . . . . . 8 |- ( ( 1st ` z ) e. RR -> ( ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) ` ( 1st ` z ) ) = ( ( ( 1st ` z ) [,) +oo ) X. RR ) )
51	45, 50	syl 17	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) ` ( 1st ` z ) ) = ( ( ( 1st ` z ) [,) +oo ) X. RR ) )
52	44, 51	eleqtrrd 2704	. . . . . 6 |- ( z e. ( RR X. RR ) -> z e. ( ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) ` ( 1st ` z ) ) )
53		elunirn2 29451	. . . . . 6 |- ( ( Fun ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) /\ z e. ( ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) ` ( 1st ` z ) ) ) -> z e. U. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) )
54	34, 52, 53	sylancr 695	. . . . 5 |- ( z e. ( RR X. RR ) -> z e. U. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) )
55	54	ssriv 3607	. . . 4 |- ( RR X. RR ) C_ U. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) )
56		ssun3 3778	. . . 4 |- ( ( RR X. RR ) C_ U. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) -> ( RR X. RR ) C_ ( U. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. U. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
57	55, 56	ax-mp 5	. . 3 |- ( RR X. RR ) C_ ( U. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. U. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
58		uniun 4456	. . 3 |- U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) = ( U. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. U. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
59	57, 58	sseqtr4i 3638	. 2 |- ( RR X. RR ) C_ U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
60	33, 59	eqssi 3619	1 |- U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) = ( RR X. RR )

Syntax hints:   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   ran crn 5115   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   [,)cico 12177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181

This theorem is referenced by:  sxbrsigalem3  30334  sxbrsigalem2  30348